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  Volume 4, Issue 2, Article 37
Inequalities Related to Rearrangements of Powers and Symmetric Polynomials

    Authors: Cezar Joita, Pantelimon Stanica,  
    Keywords: Symmetric Polynomials, Permutations, Inequalities.  
    Date Received: 29/04/03  
    Date Accepted: 19/05/03  
    Subject Codes:

05E05, 11C08, 26D05.

    Editors: Constantin P. Niculescu,  

In [2] the second author proposed to find a description (or examples) of real-valued $ n$-variable functions satisfying the following two inequalities:

if $ x_ileq y_i, i=1,ldots,n$, then $ F(x_1,ldots,x_n)leq F(y_1,ldots,y_n)$,
with strict inequality if there is an index $ i$ such that $ x_iy_i$; and for $ 0x_1x_2cdotsx_n$, then,
$displaystyle F(x_1^{x_2},x_2^{x_3},ldots,x_n^{x_1})leq F(x_1 {x_1},x_2^{x_2},cdots,x_n^{x_n}).$
In this short note we extend in a direction a result of [2] and we prove a theorem that provides a large class of examples satisfying the two inequalities, with $ F$ replaced by any symmetric polynomial with positive coefficients. Moreover, we find that the inequalities are not specific to expressions of the form $ x^y$, rather they hold for any function $ g(x,y)$ that satisfies some conditions. A simple consequence of this result is a theorem of Hardy, Littlewood and Polya [1].

[1] G. HARDY, J.E. LITTLEWOOD and G. PÓLYA, Inequalities, Cambridge Univ. Press, 2001.

[2] P. STANICA, Inequalities on linear functions and circular powers, J. Ineq. in Pure and Applied Math., 3(3) (2002), Art. 43.

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